Showing papers on "Singularity published in 2013"

Hyperbolic metamaterials

Abstract: Metamaterials with hyperbolic dispersion (where two eigenvalues of the dielectric permittivity tensor have opposite signs) exhibit a broad bandwidth singularity in the photonic density of states, with resulting manifestations in a variety of phenomena, from spontaneous emission to light propagation and scattering. In this tutorial, I will review some of the recent developments in this field.

The Witten equation, mirror symmetry, and quantum singularity theory

Abstract: For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological eld theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity Ar 1. We also resolve two outstanding conjectures of Witten. The rst conjecture is that ADE-singularities are self-dual, and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

General Argyres-Douglas Theory

Abstract: We construct a large class of Argyres-Douglas type theories by compactifying six dimensional (2, 0) A N − 1 theory on a Riemann surface with irregular singularities. We give a complete classification for the choices of Riemann surface and the singularities. The Seiberg-Witten curve and scaling dimensions of the operator spectrum are worked out. Three dimensional mirror theory and the central charges a, c are also calculated for some subsets, etc. Our results greatly enlarge the landscape of $ \mathcal{N}=2 $ superconformal field theory and in fact also include previous theories constructed using regular singularity on the sphere.

Hydrodynamic gradient expansion in gauge theory plasmas.

Abstract: We utilize the fluid-gravity duality to investigate the large order behavior of hydrodynamic gradient expansion of the dynamics of a gauge theory plasma system. This corresponds to the inclusion of dissipative terms and transport coefficients of very high order. Using the dual gravity description, we calculate numerically the form of the stress tensor for a boost-invariant flow in a hydrodynamic expansion up to terms with 240 derivatives. We observe a factorial growth of gradient contributions at large orders, which indicates a zero radius of convergence of the hydrodynamic series. Furthermore, we identify the leading singularity in the Borel transform of the hydrodynamic energy density with the lowest nonhydrodynamic excitation corresponding to a ‘nonhydrodynamic’ quasinormal mode on the gravity side.

Constraints on conformal field theories in diverse dimensions from the bootstrap mechanism.

Abstract: Recently an efficient numerical method has been developed to implement the constraints of crossing symmetry and unitarity on the operator dimensions and operator product expansion coefficients of conformal field theories in diverse space-time dimensions. It appears that the calculations can be done only for theories lying at the boundary of the allowed parameter space. Here it is pointed out that a similar method can be applied to a larger class of conformal field theories, whether unitary or not, and no free parameter remains, provided we know the fusion algebra of the low lying primary operators. As an example we calculate using first principles, with no phenomenological input, the lowest scaling dimensions of the local operators associated with the Yang-Lee edge singularity in three and four space dimensions. The edge exponents compare favorably with the latest numerical estimates. A consistency check of this approach on the 3D critical Ising model is also made.

Small resolutions of SU(5)-models in F-theory

Abstract: We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered four-folds defined by Weierstrass models admitting a split $\tilde{A}_4$ singularity over a divisor of the discriminant locus Such varieties are used to geometrically engineer SU(5) grand unified theories in F-theory The desingularization is given by a small resolution of singularities The $\tilde{A}_4$ fiber naturally appears after resolving the singularities in codimension-one in the base The remaining higher codimension singularities are then beautifully described by a four-dimensional affine binomial variety which leads to six different small resolutions of the elliptically fibered four-fold These six small resolutions define distinct four-folds connected to each other by a network of flop transitions forming a dihedral group The location of these exotic fibers in the base is mapped to conifold points of the three-folds that defines the type IIB orientifold limit of the F-theory The full resolution has interesting properties, specially for fibers in codimension-three: the rank of the singular fiber does not necessary increase and the fibers are not necessary in the list of Kodaira and some are not even (extended) Dynkin diagrams

Energy–momentum tensor from the Yang–Mills gradient flow

Abstract: The product of gauge fields generated by the Yang-Mills gradient flow for positive flow times does not exhibit the coincidence-point singularity and a local product is thus independent of the regularization. Such a local product can furthermore be expanded by renormalized local operators at zero flow time with finite coefficients that are governed by renormalization group equations. Using these facts, we derive a formula that relates the small flow-time behavior of certain gauge-invariant local products and the correctly-normalized conserved energy-momentum tensor in the Yang-Mills theory. Our formula provides a possible method to compute the correlation functions of a well-defined energy-momentum tensor by using lattice regularization and Monte Carlo simulation.

Dressing phases of AdS3/CFT2

Abstract: We determine the all-loop dressing phases of the AdS3/CFT2 integrable system related to type IIB string theory on AdS3 x S3 x T4 by solving the recently found crossing relations and studying their singularity structure. The two resulting phases present a novel structure with respect to the ones appearing in AdS5/CFT4 and AdS4/CFT3. In the strongly-coupled regime, their leading order reduces to the universal Arutyunov-Frolov-Staudacher phase as expected. We also compute their sub-leading order and compare it with recent one-loop perturbative results, and comment on their weak-coupling expansion.

Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem

Abstract: We consider the energy critical Schrodinger map problem with the 2-sphere target for equivariant initial data of homotopy index k=1. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blowup solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.

Finite time singularities for the free boundary incompressible Euler equations

Abstract: In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem) for which the smoothness of the interface breaks down in nite time into a splash singularity or a splat singularity.

Low-energy non-linear excitations in sphere packings

Abstract: We study theoretically and numerically how hard frictionless particles in random packings can rearrange. We demonstrate the existence of two distinct unstable non-linear modes of rearrangement, both associated with the opening and the closing of contacts. The first mode, whose density is characterized by some exponent θ′, corresponds to motions of particles extending throughout the entire system. The second mode, whose density is characterized by an exponent θ ≠ θ′, corresponds to the local buckling of a few particles. Extended modes are shown to yield at a much higher rate than local modes when a stress is applied. We show that the distribution of contact forces follows P(f) ∼ fmin(θ′,θ), and that imposing the restriction that the packing cannot be densified further leads to the bounds and , where γ characterizes the singularity of the pair distribution function g(r) at contact. These results extend the theoretical analysis of [Wyart, Phys. Rev. Lett., 2012, 109, 125502] where the existence of local modes was not considered. We perform numerics that support that these bounds are saturated with γ ≈ 0.38, θ ≈ 0.17 and θ′ ≈ 0.44. We measure systematically the stability of all such modes in packings, and confirm their marginal stability. The principle of marginal stability thus allows us to make clearcut predictions on the ensemble of configurations visited in these out-of-equilibrium systems, and on the contact forces and pair distribution functions. It also reveals the excitations that need to be included in a description of plasticity or flow near jamming, and suggests a new path to study two-level systems and soft spots in simple amorphous solids of repulsive particles.

Regularization of the Coulomb singularity in exact exchange by Wigner-Seitz truncated interactions: Towards chemical accuracy in nontrivial systems

Abstract: Hybrid density functionals show great promise for chemically accurate first-principles calculations, but their high computational cost limits their application in nontrivial studies, such as exploration of reaction pathways of adsorbents on periodic surfaces. One factor responsible for their increased cost is the dense Brillouin-zone sampling necessary to accurately resolve an integrable singularity in the exact exchange energy. We analyze this singularity within an intuitive formalism based on Wannier-function localization and analytically prove Wigner-Seitz truncation to be the ideal method for regularizing the Coulomb potential in the exchange kernel. We show that this method is limited only by Brillouin-zone discretization errors in the Kohn-Sham orbitals, and hence converges the exchange energy exponentially with the number of $k$ points used to sample the Brillouin zone for all but zero-temperature metallic systems. To facilitate the implementation of this method, we develop a general construction for the plane-wave Coulomb kernel truncated on the Wigner-Seitz cell in one, two, or three lattice directions. We compare several regularization methods for the exchange kernel in a variety of real systems including low-symmetry crystals and low-dimensional materials. We find that our Wigner-Seitz truncation systematically yields the best $k$-point convergence for the exchange energy of all these systems and delivers an accuracy to hybrid functionals comparable to semilocal and screened-exchange functionals at identical $k$-point sets.

Cosmology with a spin

Abstract: Using the chiral representation for spinors we present a particularly transparent way to generate the most general spinor dynamics in a theory where gravity is ruled by the Einstein-Cartan-Holst action. In such theories torsion need not vanish, but it can be reinterpreted as a four-fermion self-interaction within a torsion-free theory. The self-interaction may or may not break parity invariance, and may contribute positively or negatively to the energy density, depending on the couplings considered. We then examine cosmological models ruled by a spinorial field within this theory. We find that while there are cases for which no significant cosmological novelties emerge, the self-interaction can also turn a mass potential into an upside-down Mexican hat potential. Then, as a general rule, the model leads to cosmologies with a bounce, for which there is a maximal energy density, and where the cosmic singularity has been removed. These solutions are stable, and range from the very simple to the very complex.

The backreaction of anti-D3 branes on the Klebanov-Strassler geometry

Abstract: We present the full numerical solution for the 15-dimensional space of linearized deformations of the Klebanov-Strassler background which preserve the SU(2) × SU(2) × $ {{\mathbb{Z}}_2} $ symmetries. We identify within this space the solution corresponding to anti-D3 branes, (modulo the presence of a certain “subleading” singularity in the infrared). All the 15 integration constants of this solution are fixed in terms of the number of anti-D3 branes, and the solution differs in the UV from the supersymmetric solution into which it is supposed to decay by a mode corresponding to a rescaling of the field theory coordinates. Deciding whether two solutions that differ in the UV by a rescaling mode are dual to the same theory is involved even for supersymmetric Klebanov-Strassler solutions, and we explain in detail some of the subtleties associated to this.

Stable Blowup Dynamics for the 1‐Corotational Energy Critical Harmonic Heat Flow

Abstract: We exhibit a stable finite time blowup regime for the 1-corotational energy critical harmonic heat flow from ℝ2 into a smooth compact revolution surface of ℝ3 that reduces to the semilinear parabolic problem for a suitable class of functions f. The corresponding initial data can be chosen smooth, well localized, and arbitrarily close to the ground state harmonic map in the energy-critical topology. We give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den Berg, Hulshof, and King. Our approach lies in the continuation of the study of the 1-equivariant energy critical wave map and Schrodinger map with 2 target by Merle, Raphael, and Rodnianski. © 2012 Wiley Periodicals, Inc.

The generalized second law implies a quantum singularity theorem

Abstract: The generalized second law can be used to prove a singularity theorem, by generalizing the notion of a trapped surface to quantum situations. Like Penrose’s original singularity theorem, it implies that spacetime is null-geodesically incomplete inside black holes, and to the past of spatially infinite Friedmann–Robertson–Walker cosmologies. If space is finite instead, the generalized second law requires that there only be a finite amount of entropy producing processes in the past, unless there is a reversal of the arrow of time. In asymptotically flat spacetime, the generalized second law also rules out traversable wormholes, negative masses, and other forms of faster-than-light travel between asymptotic regions, as well as closed timelike curves. Furthermore it is impossible to form baby universes which eventually become independent of the mother universe, or to restart inflation. Since the semiclassical approximation is used only in regions with low curvature, it is argued that the results may hold in full quantum gravity. The introduction describes the second law and its time-reverse, in ordinary and generalized thermodynamics, using either the fine-grained or the coarse-grained entropy. (The fine-grained version is used in all results except those relating to the arrow of time.)

Unique continuation property and local asymptotics of solutions to fractional elliptic equations

Abstract: Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations.

Dimensional regularization of local singularities in the 4th post-Newtonian two-point-mass Hamiltonian

Abstract: The article delivers the only still unknown coefficient in the 4th post-Newtonian energy expression for binary point masses on circular orbits as a function of orbital angular frequency. Apart from a single coefficient, which is known solely numerically, all the coefficients are given as exact numbers. The shown Hamiltonian is presented in the center-of-mass frame and out of its 57 coefficients, 51 are given fully explicitly. Those coefficients are six coefficients more than previously achieved [P. Jaranowski and G. Sch\"afer, Phys. Rev. D 86, 061503(R) (2012)]. The local divergences in the point-mass model are uniquely controlled by the method of dimensional regularization. As an application, the last stable circular orbit is determined as a function of the symmetric-mass-ratio parameter.

Horizons cannot save the Landscape

Abstract: Solutions with anti-D3 branes in a Klebanov-Strassler geometry with positive charge dissolved in fluxes have a certain singularity corresponding to a diverging energy density of the RR and NS-NS three-form fluxes. There are many hopes and arguments for and against this singularity, and we attempt to settle the issue by examining whether this singularity can be cloaked by a regular event horizon. This is equivalent to the existence of asymptotically Klebanov-Tseytlin or Klebanov-Strassler black holes whose charge measured at the horizon has the opposite sign to the asymptotic charge. We find that no such KT solution exists. Furthermore, for a large class of KS black holes we considered, the charge at the horizon must also have the same sign as the asymptotic charge, and is completely determined by the temperature, the number of fractional branes and the gaugino masses of the dual gauge theory. Our result suggests that antibrane singularities in backgrounds with charge in the fluxes are unphysical, which in turn raises the question as to whether antibranes can be used to uplift AdS vacua to deSitter ones. Our results also point out to a possible instability mechanism for the antibranes.

An Efficient Inverse Kinematic Algorithm for a PUMA560-Structured Robot Manipulator

Abstract: This paper presents an efficient inverse kinematics (IK) approach which features fast computing performance for a PUMA560-structured robot manipulator. By properties of the orthogonal matrix and block matrix, the complex IK matrix equations are transformed into eight pure algebraic equations that contain the six unknown joint angle variables, which makes the solving compact without computing the reverses of the 4×4 homogeneous transformation matrices. Moreover, the appropriate combination of related equations ensures that the solutions are free of extraneous roots in the solving process, and the wrist singularity problem of the robot is also addressed. Finally, a case study is given to show the effectiveness of the proposed algorithm.

The Round Sphere Minimizes Entropy among Closed Self-Shrinkers

Abstract: The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy of the initial hypersurface bounds the entropy at all future singularities. We show here that not only does the round sphere have the lowest entropy of any closed singularity, but there is a gap to the second lowest.

Double Scaling in Tensor Models with a Quartic Interaction

Abstract: In this paper we identify and analyze in detail the subleading contributions in the 1/N expansion of random tensors, in the simple case of a quartically interacting model. The leading order for this 1/N expansion is made of graphs, called melons, which are dual to particular triangulations of the D-dimensional sphere, closely related to the "stacked" triangulations. For D<6 the subleading behavior is governed by a larger family of graphs, hereafter called cherry trees, which are also dual to the D-dimensional sphere. They can be resummed explicitly through a double scaling limit. In sharp contrast with random matrix models, this double scaling limit is stable. Apart from its unexpected upper critical dimension 6, it displays a singularity at fixed distance from the origin and is clearly the first step in a richer set of yet to be discovered multi-scaling limits.

The vlasov–poisson–boltzmann system for soft potentials

Abstract: An important physical model describing the dynamics of dilute weakly ionized plasmas in the collisional kinetic theory is the Vlasov–Poisson–Boltzmann system for which the plasma responds strongly to the self-consistent electrostatic force. This paper is concerned with the electron dynamics of kinetic plasmas in the whole space when the positive charged ion flow provides a spatially uniform background. We establish the global existence and optimal convergence rates of solutions near a global Maxwellian to the Cauchy problem on the Vlasov–Poisson–Boltzmann system for angular cutoff soft potentials with -2 ≤ γ < 0. The main idea is to introduce a time-dependent weight function in the velocity variable to capture the singularity of the cross-section at zero relative velocity.

Isosingular Sets and Deflation

Abstract: This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations constructed by taking the closure of points with a common singularity structure The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry We introduce a strong form of deflation and define deflation sequences, which, in a different context, are the sequences arising in Thom---Boardman singularity theory We then define isosingular sets in terms of deflation sequences We also define the isosingular local dimension and examine the properties of isosingular sets While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic set that can be investigated with methods from numerical algebraic geometry

Double scaling in tensor models with a quartic interaction

Abstract: In this paper we identify and analyze in detail the subleading contributions in the 1/N expansion of random tensors, in the simple case of a quartically interacting model. The leading order for this 1/N expansion is made of graphs, called melons, which are dual to particular triangulations of the D-dimensional sphere, closely related to the “stacked” triangulations. For D < 6 the subleading behavior is governed by a larger family of graphs, hereafter called cherry trees, which are also dual to the D-dimensional sphere. They can be resummed explicitly through a double scaling limit. In sharp contrast with random matrix models, this double scaling limit is stable. Apart from its unexpected upper critical dimension 6, it displays a singularity at fixed distance from the origin and is clearly the first step in a richer set of yet to be discovered multi-scaling limits.

A clarification of the role of crack-tip conditions in linear elasticity with surface effects:

Abstract: We examine the role of crack-tip conditions in the reduction of stress at a crack tip in a theory of linear elasticity with surface effects. The maximum number of allowable end conditions for complete removal of a stress singularity is demonstrated for both plane and anti-plane problems. In particular, we show that the necessary and sufficient conditions for bounded stresses at a crack tip cannot be satisfied with a first-order (curvature-independent) theory of surface effects, which leads, at most, to the reduction of the classical strong square-root singularity to a weaker logarithmic singularity.